Group #28502996 — subgroups

Subgroups — #28502996

Order 15104.

218 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21426249	C₂	1	2	yes	maximal
2	#21423714	C₂	1	1	yes	maximal, characteristic
3	C₈×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
4	#21422153	C₂	1	2	yes	maximal
5	C₅₉×#1370	C₂	1	1	yes	maximal, characteristic
6	#16837807	—	2	1	no	—
7	#16837807	C₂²	1	2	yes	—
8	#16837782	—	2	1	no	—
9	C₈×Dic₁₁₈	—	2	1	no	—
10	C₄×Dic₂₃₆	C₂²	1	1	yes	characteristic
11	#16837228	—	2	2	no	—
12	#16837207	—	2	2	no	—
13	#16837155	C₂²	1	1	yes	characteristic
14	#16837134	—	2	1	no	—
15	C₅₉×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₈²×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₃₆	—	4	2	no	—
18	C₂×Dic₂₃₆	D₄	1	1	yes	characteristic
19	#14273727	—	2	2	no	—
20	C₄×Dic₁₁₈	—	2	1	no	—
21	#14273661	—	2	2	no	—
22	#14273661	—	2	2	no	—
23	#14273661	—	4	1	no	—
24	#14273661	—	4	1	no	—
25	#14273661	D₄	1	1	yes	characteristic
26	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
27	Dic₅₉⋊C₈	—	4	1	no	—
28	#14273652	—	2	1	no	—
29	C₈×Dic₅₉	—	4	1	no	—
30	C₂×C₁₆×C₅₉	—	2	2	no	—
31	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
32	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
33	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
34	C₂×Dic₁₁₈	—	2	1	no	—
35	C₂×Dic₁₁₈	—	4	2	no	—
36	C₂×C₈×C₅₉	—	2	1	no	—
37	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
38	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
39	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
41	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
42	C₄⋊Dic₅₉	—	2	1	no	—
43	C₄⋊Dic₅₉	—	4	2	no	—
44	C₄⋊Dic₅₉	—	4	2	no	—
45	Dic₅₉⋊C₄	—	4	1	no	—
46	C₄×Dic₅₉	—	4	1	no	—
47	C₂×C₅₉⋊C₈	—	4	1	no	—
48	Dic₂₃₆	—	2	1	no	—
49	Dic₂₃₆	—	8	4	no	—
50	Dic₂₃₆	Q₁₆	1	2	yes	complemented
51	C₁₆×C₅₉	—	4	2	no	—
52	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
53	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×Dic₅₉	—	4	1	no	—
56	C₂×Dic₅₉	—	4	1	no	—
57	C₂×Dic₅₉	—	8	2	no	—
58	Dic₁₁₈	—	2	2	no	—
59	Dic₁₁₈	—	4	1	no	—
60	Dic₁₁₈	—	8	4	no	—
61	C₈×C₅₉	—	2	2	no	—
62	C₈×C₅₉	—	2	1	no	—
63	C₈×C₅₉	—	2	1	no	—
64	C₈×C₅₉	Q₃₂	1	2	yes	—
65	C₈×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
66	C₈×C₅₉	#135	1	1	yes	characteristic
67	C₅₉⋊C₈	—	8	1	no	—
68	#12448	—	59	1	no	maximal, Sylow
69	C₂²×C₅₉	#459	1	1	yes	characteristic
70	C₄×C₅₉	—	2	1	no	—
71	C₄×C₅₉	#467	1	1	yes	characteristic
72	C₄×C₅₉	#473	1	1	yes	characteristic
73	C₄×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
74	C₄×C₅₉	#510	1	1	yes	characteristic
75	Dic₅₉	—	4	2	no	—
76	Dic₅₉	—	8	1	no	—
77	Dic₅₉	—	16	2	no	—
78	#2227	—	59	2	no	—
79	#1713	—	59	1	no	—
80	C₈×Q₁₆	—	59	1	no	—
81	#1370	D₅₉	1	1	yes	characteristic
82	#1365	—	59	2	no	—
83	C₂×C₅₉	#1669	1	1	yes	characteristic
84	C₂×C₅₉	#2209	1	1	yes	characteristic
85	C₂×C₅₉	#2213	1	1	yes	characteristic
86	#500	—	59	2	no	—
87	#500	—	118	1	no	—
88	#494	—	118	1	no	—
89	C₈×Q₈	—	118	1	no	—
90	C₄×Q₁₆	—	59	1	no	—
91	#366	—	118	2	no	—
92	C₄⋊C₁₆	C₂×D₅₉	1	2	yes	—
93	#358	—	118	2	no	—
94	#335	—	59	1	no	—
95	Q₈⋊C₈	—	118	1	no	—
96	C₈²	C₂×D₅₉	1	1	yes	characteristic
97	C₅₉	#12448	1	1	yes	characteristic, complemented, Sylow
98	C₂×Q₁₆	—	59	1	no	—
99	C₂×Q₁₆	—	236	2	no	—
100	C₄⋊Q₈	—	118	2	no	—
101	C₄×Q₈	—	118	1	no	—
102	C₂×C₁₆	—	2	2	no	—
103	#107	—	59	1	no	—
104	#107	—	118	2	no	—
105	#107	—	118	2	no	—
106	#107	—	236	1	no	—
107	#107	—	236	1	no	—
108	C₄⋊C₈	—	118	1	no	—
109	C₄⋊C₈	—	236	1	no	—
110	#103	—	118	1	no	—
111	C₄×C₈	—	236	1	no	—
112	C₄×C₈	D₂₃₆	1	1	yes	characteristic
113	C₄×C₈	D₂₃₆	1	1	yes	characteristic
114	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
115	C₂×Q₈	—	118	1	no	—
116	C₂×Q₈	—	236	2	no	—
117	Q₁₆	—	59	2	no	—
118	Q₁₆	—	118	1	no	—
119	Q₁₆	—	472	4	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	236	1	no	—
122	C₂×C₈	D₄₇₂	1	1	yes	characteristic
123	C₂×C₈	D₄₇₂	1	1	yes	characteristic
124	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
125	C₂×C₈	#11745689	1	1	yes	characteristic
126	C₄⋊C₄	—	118	1	no	—
127	C₄⋊C₄	—	236	2	no	—
128	C₄⋊C₄	—	236	2	no	—
129	C₄⋊C₄	—	236	1	no	—
130	C₄²	—	236	1	no	—
131	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
132	C₁₆	—	4	2	no	—
133	Q₈	—	118	2	no	—
134	Q₈	—	472	4	no	—
135	Q₈	—	236	1	no	—
136	C₂×C₄	—	472	2	no	—
137	C₂×C₄	—	236	1	no	—
138	C₂×C₄	—	236	1	no	—
139	C₂×C₄	#14273702	1	1	yes	characteristic
140	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
141	C₂×C₄	#14273755	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	472	1	no	—
146	C₈	Dic₄₇₂	1	2	yes	—
147	C₈	Q₁₆×D₅₉	1	1	yes	characteristic
148	C₈	#14273746	1	1	yes	characteristic
149	C₂²	#16837598	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	944	2	no	—
152	C₄	—	472	1	no	—
153	C₄	—	236	2	no	—
154	C₄	#16837644	1	1	yes	characteristic
155	C₄	#16837678	1	1	yes	characteristic
156	C₄	C₂×Dic₄₇₂	1	1	yes	characteristic
157	C₄	#16837845	1	1	yes	characteristic
158	C₂	#21423462	1	1	yes	characteristic
159	C₂	#21426147	1	1	yes	characteristic
160	C₂	#21426173	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21426249

C₂

yes

maximal

#21423714

C₂

yes

maximal, characteristic

C₈×Dic₂₃₆

C₂

yes

maximal, characteristic

#21422153

C₂

yes

maximal

C₅₉×#1370

C₂

yes

maximal, characteristic

#16837807

—

#16837807

C₂²

yes

—

#16837782

—

C₈×Dic₁₁₈

—

C₄×Dic₂₃₆

C₂²

yes

—

—

C₂²

yes

characteristic

#16837134

—

C₅₉×C₄⋊C₁₆

C₂²

yes

—

C₈²×C₅₉

C₂²

yes

characteristic

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

D₄

yes

characteristic

#14273727

—

C₄×Dic₁₁₈

—

—

—

—

—

D₄

yes

characteristic

C₄⋊(C₅₉⋊C₈)

—

Dic₅₉⋊C₈

—

#14273652

—

C₈×Dic₅₉

—

C₂×C₁₆×C₅₉

—

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₄×C₈×C₅₉

C₂³

yes

characteristic

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

D₈

yes

characteristic

C₂×C₈×C₅₉

D₈

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

#41

yes

characteristic

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₂×C₅₉⋊C₈

—

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

Q₁₆

yes

complemented

C₁₆×C₅₉

—

C₂×C₄×C₅₉

#121

yes

characteristic

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

Q₃₂

yes

—

C₈×C₅₉

C₂×Q₁₆

yes

characteristic

C₈×C₅₉

#135

yes

characteristic

C₅₉⋊C₈

—

—

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

C₂×Q₃₂

yes

characteristic

C₄×C₅₉

#510

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

#2227

—

#1713

—

C₈×Q₁₆

—

#1370

D₅₉

yes

characteristic

#1365

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

118

—

#494

—

118

—

C₈×Q₈

—

118

—

C₄×Q₁₆

—

#366

—

118

—

C₄⋊C₁₆

C₂×D₅₉

yes

—

#358

—

118

—

#335

—

Q₈⋊C₈

—

118

—

C₈²

C₂×D₅₉

yes

characteristic

C₅₉

#12448

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

236

—

100

C₄⋊Q₈

—

118

—

101

C₄×Q₈

—

118

—

102

C₂×C₁₆

—

103

#107

—

104

#107

—

118

—

105

#107

—

118

—

106

#107

—

236

—

107

#107

—

236

—

108

C₄⋊C₈

—

118

—

109

C₄⋊C₈

—

236

—

110

#103

—

118

—

111

C₄×C₈

—

236

—

112

C₄×C₈

D₂₃₆

yes

characteristic

113

C₄×C₈

D₂₃₆

yes

characteristic

114

C₄×C₈

C₂²×D₅₉

yes

characteristic

115

C₂×Q₈

—

118

—

116

C₂×Q₈

—

236

—

117

Q₁₆

—

118

Q₁₆

—

118

—

119

Q₁₆

—

472

—

120

C₂×C₈

—

121

C₂×C₈

—

236

—

122

C₂×C₈

D₄₇₂

yes

characteristic

123

C₂×C₈

D₄₇₂

yes

characteristic

124

C₂×C₈

D₄×D₅₉

yes

characteristic

125

C₂×C₈

#11745689

yes

characteristic

126

C₄⋊C₄

—

118

—

127

C₄⋊C₄

—

236

—

128

C₄⋊C₄

—

236

—

129

C₄⋊C₄

—

236

—

130

C₄²

—

236

—

131

C₄²

C₂×D₂₃₆

yes

characteristic

132

C₁₆

—

133

Q₈

—

118

—

134

Q₈

—

472

—

135

Q₈

—

236

—

136

C₂×C₄

—

472

—

137

C₂×C₄

—

236

—

138

C₂×C₄

—

236

—

139

C₂×C₄

#14273702

yes

characteristic

140

C₂×C₄

C₂×D₄₇₂

yes

characteristic

141

C₂×C₄

#14273755

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

472

—

146

C₈

Dic₄₇₂

yes

—

147

C₈

Q₁₆×D₅₉

yes

characteristic

148

C₈

#14273746

yes

characteristic

149

C₂²

#16837598

yes

characteristic

150

C₄

—

151

C₄

—

944

—

152

C₄

—

472

—

153

C₄

—

236

—

154

C₄

#16837644

yes

characteristic

155

C₄

#16837678

yes

characteristic

156

C₄

C₂×Dic₄₇₂

yes

characteristic

157

C₄

#16837845

yes

characteristic

158

C₂

#21423462

yes

characteristic

159

C₂

#21426147

yes

characteristic

160

C₂

#21426173

yes

characteristic

	subgroup	G/H
center	C₂²	#16837598
commutator, cosocle	C₄×C₈×C₅₉	C₂³
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#1370	C₂
socle, prime radical	C₂²×C₅₉	#459
hypercenter	#1370	D₅₉
hypocenter	C₅₉	#12448
LCS	G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1370

subgroup

G/H

center

C₂²

#16837598

commutator, cosocle

C₄×C₈×C₅₉

C₂³

Frattini

C₄×C₈

C₂²×D₅₉

Fitting

C₅₉×#1370

C₂

C₂²×C₅₉

D₅₉

C₅₉

G ▷ C₄×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1370

N	H ≅ G/N	type
50. Dic₂₃₆	Q₁₆	semidirect
97. C₅₉	#12448	semidirect

H ≅ G/N

type

50. Dic₂₃₆

Q₁₆

semidirect

97. C₅₉

#12448

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#21426249	2
#21423714	2
C₈×Dic₂₃₆	2
#21422153	2
C₅₉×#1370	2
#12448	59

subgroup

index

C₈×Dic₂₃₆

C₅₉×#1370

Subgroups — #28502996

Conjugacy classes of subgroups

Complements

Maximal subgroups